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A converse of Berndtsson's theorem on the positivity of direct images

Wang Xu, Hui Yang

TL;DR

This work establishes an intrinsic converse to Berndtsson's positivity of direct images in the setting of a projective fibration $p:X\to Y$. It shows that if, for every semi-positive line bundle $E\to X$, the direct image $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive, then the curvature of $(L,h_L)$ must be semi-positive, with the proof leveraging localization and the relative Bergman kernel framework. A key methodological feature is the construction of auxiliary data on $X$ to translate geometric positivity into analytic constraints that yield a contradiction unless $L$ is semi-positive. Consequently, Griffiths semi-positivity of the family of direct images implies Nakano semi-positivity via Berndtsson's theorem, strengthening the link between fiberwise positivity and base positivity in a cohesive L^2/complex Brunn–Minkowski context.

Abstract

Berndtsson's famous theorem asserts that, for a compact Kähler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective fibration: if $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle $E\to X$, then the curvature of $L$ must be semi-positive.

A converse of Berndtsson's theorem on the positivity of direct images

TL;DR

This work establishes an intrinsic converse to Berndtsson's positivity of direct images in the setting of a projective fibration . It shows that if, for every semi-positive line bundle , the direct image is Griffiths semi-positive, then the curvature of must be semi-positive, with the proof leveraging localization and the relative Bergman kernel framework. A key methodological feature is the construction of auxiliary data on to translate geometric positivity into analytic constraints that yield a contradiction unless is semi-positive. Consequently, Griffiths semi-positivity of the family of direct images implies Nakano semi-positivity via Berndtsson's theorem, strengthening the link between fiberwise positivity and base positivity in a cohesive L^2/complex Brunn–Minkowski context.

Abstract

Berndtsson's famous theorem asserts that, for a compact Kähler fibration , the direct image bundle of a semi-positive Hermitian holomorphic line bundle is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective fibration: if is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle , then the curvature of must be semi-positive.
Paper Structure (3 sections, 6 theorems, 73 equations, 1 figure)

This paper contains 3 sections, 6 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Proof of Theorem \ref{['MainThm']}

Key Result

Theorem 1.1

Let $p:X\to Y$ be a proper holomorphic submersion from a Kähler manifold $X$ to a connected complex manifold $Y$. Let $(L,h)$ be a semi-positive Hermitian holomorphic line bundle over $X$. Then the induced $L^2$ metric on the direct image bundle $p_*(K_{X/Y}\otimes L)$ is Nakano semi-positive.

Figures (1)

  • Figure :

Theorems & Definitions (6)

  • Theorem 1.1: Berndtsson Berndtsson09
  • Theorem 1.2: Li-Xu-Zhou LiXuZhou
  • Theorem 1.3: Xu-Yang XuYang
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 3.1